Adaptive control for memristive system via compensatory controller and Chebyshev neural network

In this paper, based on linear matrix inequality technique, a simple controller and a compensatory controller are designed. It can track arbitrary fixed points and any periodic orbits. In addition, a synchronization control method via Chebyshev neural network with external disturbances is proposed. An adaptive controller is given. The Chebyshev neural network is used to approximate the uncertain nonlinear function and the adaptive law is used to adjust the corresponding parameters in the system. Taking a 4D memristive chaotic system as examples, the results are in consistent with the simulations. From a framework and control theoretical point of view, the proposed synchronization approach via compensation controller and Chebyshev neural network is firstly presented. From an application point of view, the proposed scheme can simplify the complexity of controller design. It is promising in many applications for mem-systems as secure communications and neural networks.

The brain is composed of neurons with different structures and functions, and studying the structure and function of these neurons is a significant and challenging frontier hot spot 1 .Through the continuous exploration of scholars, the neural network has been formed to study brain science based on the structure and function of neurons 2 .Neural network simulates neuronal activities with mathematical models, which can realize human brain's thinking, perception and other behaviors 3 .In the past few decades, neural networks have received extensive attention from scholars and have been applied in many fields, such as secure communication 4 , fault detection 5 , intelligent control 6 , machine learning 7 and deep learning 8 .In the general neural network model, the resistors act as biological synapses, but ordinary resistors can not meet the needs of memory, and memristors can achieve this function.The memristor was proposed by Professor Chua 9 in 1971 and implemented by Strukov and Snidein 10 in 2008.Memristor neural networks are characterized by plastic response, non-volatility, nanoscale size, low energy consumption, storability and easy integration 11 .In addition, among the many dynamic behaviors of neural networks, tracking and synchronization are very meaningful research content 12 .Tracking and synchronization mean that related neural network nodes achieve the same dynamic behavior.Pinning synchronization can be divided into pinning synchronization 13 , sliding mode synchronization 14 and exponential synchronization 15 and fixed/prescribed-time synchronization 16 , etc.However, it is often impossible to obtain the exact initial state value of the system in practical application.In order to solve this problem, Cheng and Tang 17 proposed the finite-time and fixed-time synchronization of delayed memory neural networks based on adaptive aperiodic intermittent adjustment strategy in 2023.Compared with the uncertainty of infinite rest time in asymptotic and exponential synchronization theory and the initial value dependence of finite rest time in finite time synchronization theory, the certainty and independence of finite and fixed-time synchronization theory have been greatly improved.The practical applications of neural network tracking and synchronization theory, such as tracking control and secure communication, have received extensive attention and in-depth research [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] .
However, for the above works, the controller design is only for a specific chaotic networks and does not have universality.The adaptive control of memristive system based on linear matrix inequalities technique and Chebyshev neural network has rarely been reported.
Based on Lyapunov stability theory, a suitable nonlinear adaptive controller and compensation controller are designed in this paper, and the synchronization problem of a class of memristive systems and neural networks is studied.The main highlights and contributions are summarized as follows: (1) Different from the traditional stability theory, the compensatory controller and adaptive controller used in this paper require fewer parameters, reduce energy consumption, and have better economic mechanism.(2) Adding memristive components that are highly similar to biological synapses to the traditional neural network model can better simulate the human brain.(3) Based on linear matrix inequality technique, a simple controller and a compensatory controller and the update law are designed.(4) A novel synchronization method via Chebyshev neural network is proposed, then, taking this memristve system with external disturbances as an example, the simulation results verify the effectiveness of the proposed method.

Tracking control method
The memristive system is described as where x ∈ R n is the state variable, A ∈ R n×n is the coefficient matrix, F(x) ∈ R n is the nonlinear function which satisfies the following Lipschitz conditions: The structure of the proposed adaptive controlled memristive system is shown in Fig. 1.Assuming the reference signal x = [x 1 , x2 , . . ., xn ] T and a tracking controller U to (1), one has In order to realize x to track the reference signal, it needs the next steps.Applying a compensatory controller û = [û 1 , û2 , . . ., ûn ] T to (1), one obtains Defining the error where u is the adaptive controller, Subtracting (3) from ( 4).One has Theorem 1 Under the nonlinear function F(x) − F(x) of (6) satisfies the Lipschitz condition (2), if the controller and update law are designed, respectively, as where q ∈ R + , µ ∈ R + , Ŵ ∈ R n×n , ∈ R n×n is a positive definite symmetric matrix which satisfies the following linear matrix inequality : (1)

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The error e(t) is globally asymptotically stable to zero under the action of the controller (7) and update law (8).

Proof Define Lyapunov function as
Since is a positive definite matrix, V is also positive definite.One has

One has
According to (9), one has Therefore, the error ( 6) is globally asymptotically stable at zero.The proof is completed.

Remark 1
The linear matrix inequality Eq. ( 9) holds that all the eigenvalues of the matrix A − Ŵ are negative.And the controller u is derived by selecting a matrix such that all eigenvalues of the matrix A − Ŵ are negative, and a positive definite symmetric matrix is arbitrarily defined.The positive definite symmetric matrix can be obtained, so that the adaptive controller u can be determined by (7).The expression of the compensatory controller û can be easily determined by substituting the reference signal into the system.

Synchronization control via Chebyshev neural network
The driven system is described as The corresponding response system is given as And φ(x) can be written as www.nature.com/scientificreports/ The controller can be designed as The compensatory controller is designed as where σ > 0 is the matrix of feedback gains, is a symmetric positive definite matrix as (9). is a constant.M = I , f = T η is the uncertain estimator using Chebyshev neural network, suppose that the first five polynomials are chosen as follow: where ] , ηi = [w 0i , w 1i , w 2i , w 3i , w 4i ] T , and w 0i , w 1i , …,w 4i denote the coefficients of Chebyshev neural network to be estimated.

Proof Define Lyapunov function as
where is a symmetric positive definite matrix as (9).η = η * − η,γ is a positive constant.The time derivative of V is One has where �ϑ(t)� ≤ , let Then, Eq. ( 26) can be rewritten as

Results and discussion
Taking the memristive system (34) as an example, rewrite it as , according to (6), the compensatory controller can be obtained as Let Ŵ = Diag10, 0, 0, 10 , one has The eigenvalues of the matrix A − Ŵ can be obtained as follow It can be seen that all the eigenvalues are negative, and the positive real symmetric matrix = Diag (5, 5, 5, 5) is used, according to linear matrix inequality (12), a positive symmetric matrix can be gotten as According to Eqs. ( 7), ( 8) and ( 9), when the time domain response and the adaptive parameter k(t) for the controlled mem-system (5) are drawn in Fig. 4.

Conclusion
To sum up, a memristive system and the basic principle of tracking control method were introduced, and the stability theory about the tracking control was analyzed.The process was to track the signal with the chaotic variable information driven by the memristive system.Additionally, a synchronization method via Chebyshev neural network was proposed.Compared to previous control schemes, in this scheme, it has the following achievements: (1) The adaptive control methods of memristive system based on linear matrix inequality and Chebyshev neural network are firstly proposed.(2) It has wide control range.This scheme can stabilize the memristive system at arbitrary fixed points or any periodic orbit (see Figs. 4 and 5), the synchronization of two identical mem-systems with external disturbances can be fast achieved (see Fig. 6).The control process is simple and effective.Moreover, the structure of Chebyshev neural networks is simpler since there are fewer tuning parameters. it is particularly easy to implement in engineering.The fractional-order control method and their application in communication will be discussed in next work.

Figure 1 .
Figure 1.The structure of the proposed adaptive controlled system.

2 . 1 Theorem 2
and B are constant matrice, A and B are the perturba- tions, f and g are nonlinear functions, f and g are uncertainties, d m (t) and d s (t) are external disturbances and the synchronization error e = y − x .One has The structure of Chebyshev neural network is depicted as Fig. Lemma If lim t→∞ ξ(t) → � and ξ(t) is bounded, then, lim t→∞ ξ(t) → 0. Considering the interval [−1, 1] , Chebyshev polynomials, which are given as and the function φ(x) of Chebyshev neural network is considered as where W = [w 0 , w 1 , . . ., w m ] T , T = [T 0 , T 1 , . . ., T m ] T .

Figure 2 .
Figure 2. The structure of Chebyshev neural network.

1 =Figure 4 .
Figure 4.The tracking control of fixed-point P .(a) Time domain response, (b) the parameter k(t).

Figure 5 .
Figure 5.The tracking control of periodic signal, (a) time domain response, (b) the parameter k(t).